Problem 13 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \(y=5^{\sqrt{s}}\)

Step-by-Step Solution

Answer

\(\frac{dy}{ds} = \frac{\ln(5) \cdot 5^{\sqrt{s}}}{2\sqrt{s}}\)

Step 1: Identify the Differentiation Rule

To differentiate the function \(y = 5^{\sqrt{s}}\), we need to use the chain rule and the exponential differentiation rule. Specifically, \(y = a^{u}\) differentiates with respect to \(x\) as \(y' = a^{u} \ln(a) \cdot \frac{du}{dx}\), where \(u\) is a differentiable function of \(x\).

Step 2: Differentiate the Inner Function

Set \(u = \sqrt{s}\). Now find the derivative of \(u\) with respect to \(s\), which is \(\frac{du}{ds} = \frac{d}{ds}(s^{1/2}) = \frac{1}{2}s^{-1/2} = \frac{1}{2\sqrt{s}}\).

Step 3: Apply the Exponential Rule

Using the exponential differentiation rule, \(\frac{dy}{ds} = 5^{\sqrt{s}} \ln(5) \cdot \frac{du}{ds}\).

Step 4: Substitute and Simplify

Substitute \(\frac{du}{ds}\) from Step 2 into the expression from Step 3, resulting in: \(\frac{dy}{ds} = 5^{\sqrt{s}} \cdot \ln(5) \cdot \frac{1}{2\sqrt{s}} = \frac{\ln(5) \cdot 5^{\sqrt{s}}}{2\sqrt{s}}\).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. In simple terms, it helps us differentiate functions nested within other functions. Imagine having one function inside another; the chain rule tells us how fast the output of the innermost function changes as the input changes.

When you see a function like \[ y = f(g(x)) \]where one function is nested inside another, the derivative of this composite function is calculated using the chain rule as \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]where:

• \( f'(g(x)) \) represents the derivative of the outer function evaluated at the inner function.
• \( g'(x) \) is the derivative of the inner function.

In our specific case with \( y = 5^{\sqrt{s}} \),we first identify the inner function \( u = \sqrt{s} \). To find the full derivative, differentiate the outer and inner functions separately and then multiply their derivatives accordingly.

Remember, understanding the chain rule is crucial as it frequently appears in calculus problems involving functions within functions.

Exponential Differentiation

Exponential differentiation refers to the process of finding the derivative of a function where the variable is in the exponent. This type of differentiation is significant because it shows up in many real-world applications, especially in fields like biology and finance.

When differentiating an exponential function such as \( y = a^u \),we use the formula:\[ \frac{dy}{dx} = a^u \ln(a) \cdot \frac{du}{dx} \]where:

• \( a^u \) is an exponential function with base \( a \) and exponent \( u \).
• \( \ln(a) \) is the natural logarithm of the base \( a \).
• \( \frac{du}{dx} \) represents the derivative of the exponent \( u \) with respect to \( x \).

In our exercise, the base is 5and the exponent is \( \sqrt{s} \). We first find the derivative of the exponent and then plug it into our differentiation formula. This specific rule helps simplify complex-looking equations that contain exponential terms, guiding us to a clean and correct derivative result.

Differentiation Rules

Differentiation rules are indispensable tools in calculus. They provide formulas and techniques to easily find derivatives of various functions. By understanding these rules, we can quickly identify how to handle different kinds of functions.

Some of the most used differentiation rules include:

• Power Rule: When the function is of the form \( x^n \), the derivative is \( nx^{n-1} \).
• Product Rule: For finding the derivative of products of two functions.
• Quotient Rule: Useful for terms that are ratios of two functions.
• Chain Rule: A must-know for functions within functions.
• Exponential Rule: Essential for functions where the variable is in the exponent.

In the original problem, we primarily employed the chain rule and the exponential rule. Mastery of differentiation rules means you can tackle almost any differentiation problem as they lay down the basic roadmap for tackling derivatives effectively. Learning each rule and practicing their applications provides the confidence needed to approach more complex calculus problems.